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Multiple Choice Let \(C\) denote a curve represented by the parametric equations \(x=x\,(t),\quad \ y=y(t),\quad \ a\leq t\leq b\), where each function \(x(t) \,\) and \(y(t) \) is continuous on the closed interval \([ a,b] \) and differentiable on the open interval \(( a,b) \). If both \(\dfrac{dx}{dt}\) and \(\dfrac{dy}{dt}\) are continuous and never simultaneously \(0\) on \(( a,b) \), then \(C\) is called a [(a) smooth, (b) differentiable, (c) parametric] curve.

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True or False If \(C\) is a smooth curve, represented by the parametric equations \(x=x(t)\) and \(y=y(t),\) \(a\leq t\leq b\), then the slope of the tangent line to \(C\) at the point \(( x, y) \) is given by the formula \(\dfrac{dy}{dx}=\dfrac{\dfrac{dy}{dt} }{\dfrac{dx}{dt}},\) provided \(\dfrac{dx}{dt}\neq 0\).

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If in the formula for the slope of a tangent line, \(\dfrac{dy}{dx}=0\) \(\bigg(but \dfrac{dx}{dt}\neq 0\bigg),\) then the curve has a(n) ______________ tangent line at the point \(( x(t) ,y(t) ) .\) If \(\dfrac{dx}{dt}=0\) \(\bigg(but \dfrac{dy}{dt}\neq 0 \bigg)\), then the curve has a(n) ______________ tangent line at the point \(( x(t) ,y(t) ) .\)

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True or False For a smooth curve \(C\) represented by the parametric equations \(x=x(t),\quad \ y=y(t),\) \(a\leq t\leq b,\) the length \(s\) of \(C\) from \(t=a\) to \(t=b\) is given by the formula \(s=\int_{a}^{b}\sqrt{ \dfrac{d^{2}x}{dt^{2}}+\dfrac{d^{2}y}{dt^{2}}}~dt\).

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\(x(t) =e^{t}\cos t,\;y(t) =e^{t}\sin t\)

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\(x(t) =1+e^{-t},\;y(t) =e^{3t}\)

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\(x(t) =t+\dfrac{1}{t},\;y(t) =4+t\)

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\(x(t) =t+\dfrac{1}{t},\;y(t) =t-\dfrac{1}{t}\)

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\(x(t) = \cos t+t\sin t,\;y(t) = \sin t-t\cos t\)

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\(x(t) =\cos ^{3}t,\;y(t) =\sin ^{3}t\)

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\(x(t) =\cot ^{2}t,\;y(t) =\cot t\)

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\(x(t) =\sin t,\;y(t) =\sec ^{2}t\)

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\(x(t) =2t^{2},\quad y(t) =t\quad at\quad t=2\)

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\(x(t) =t,\quad y(t) =3t^{2}\quad at\quad t=-2\)

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\(x(t) =3t,\quad y(t) =2t^{2}-1\quad at\quad t=1\)

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\(x(t) =2t,\quad y(t) =t^{2}-2\quad at\;t=2\)

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\(x(t) =\sqrt{t},\quad y(t) =\dfrac{1}{t}\quad at\;t=4\)

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\(x(t) =\dfrac{2}{t^{2}},\quad y(t) =\dfrac{1}{t}\quad at\;t=1\)

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\(x(t) =\dfrac{t}{t+2},\quad y(t) =\dfrac{4}{t+2}\quad at\;t=0\)

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\(x(t) =\dfrac{t^{2}}{1+t},\quad y(t) =\dfrac{1}{1+t}\quad at\;t=0\)

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\(x(t) =e^{t},\quad y(t) =e^{-t}\quad at\;t=0\)

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\(x(t) =e^{2t},\quad y(t) =e^{t}\quad at\;t=0\)

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\(x(t) =\sin t,\quad y(t) =\cos t\quad at\;t=\dfrac{\pi }{4}\)

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\(x(t) =\sin ^{2}t,\quad y(t) =\cos t\quad at\;t=\dfrac{\pi }{4}\)

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\(x(t) =4\sin t,\quad y(t) =3\cos t\quad at\;t=\dfrac{\pi }{3}\)

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\(x(t) =2\sin t-1,\quad y(t) =\cos t+2\quad at\;t=\dfrac{\pi }{6}\)

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\(x(t)=t^2,\quad y(t)=t^3-4t\)

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\(x(t) = t^3 -9t ,\quad y(t)=t^2\)

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\(x(t) =1-\cos t,\quad y(t) =1-\sin t,\quad 0\leq t \leq 2\pi\)

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\(x(t) =-3\cos t+\cos (3t),\quad y(t) =\sin t,\quad 0\leq t\leq 2\pi\)

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\(x(t)=t^{3},\quad y(t)=t^{2};\quad 0\leq t\leq 2\)

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\(x(t)=3t^{2}+1,\quad y(t)=t^{3}-1;\quad 0\leq t\leq 2\)

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\(x(t)=t-1,\quad \ y(t)=\dfrac{1}{2}t^{2};\quad 0\leq t\leq 2\)

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\(x(t)=t^{2},\quad y(t)=2t;\;1\leq t\leq 3 \)

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\(x(t)=4\sin t,\quad y(t)=4\cos t;\; -\dfrac{\pi }{2}\leq t\leq \dfrac{\pi }{2}\)

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\(x(t)=6\sin t,\quad y(t)=6\cos t;\; -\dfrac{\pi }{2}\leq t\leq \dfrac{\pi }{2}\)

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\(x(t)=2\sin t-1,\quad y(t)=2\cos t+1;\quad 0\leq t\leq 2\pi \)

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\(x(t)=e^{t}\sin t,\quad y(t)=e^{t}\cos t;\quad 0\leq t\leq \pi \)

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\(x(t) =2\cos ( 2t) ,\quad y(t) =t^{2};\;0\leq t\leq 2\pi \)

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\(x(t) =t^{2},\quad y(t) =\sin t;\quad 0\leq t\leq 2\pi \)

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\(x(t) =t^{2},\quad y(t) =\sqrt{t+2};\;-2\leq t\leq 2\)

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\(x(t) =t-\cos t,\quad y(t) =1-\sin t;\quad 0\leq t\leq \pi \)

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\(x(t) =3\cos t+\cos \left( 3t\right) ,\quad y(t) =3\sin t-\sin ( 3t);\;0\leq t\leq 2\pi \) (a hypocycloid)

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\(x(t) =5\cos t-\cos ( 5t) ,\quad y(t) =5\sin t-\sin ( 5t) ;\; 0\leq t\leq 2\pi \) (an epicycloid)

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Tangent Lines

1. Find all the points on the plane curve \(C\) represented by \(x(t) =t^{2}+2,\quad y(t) =t^{3}-4t\), where the tangent line is horizontal or where it is vertical.
2. Show that \(C\) has two tangent lines at the point \(( 6,0) .\)
3. Find equations of these tangent lines.
4. Graph \(C\).

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Tangent Lines

1. Find all the points on the plane curve \(C\) represented by \(x(t) =2t^{2},\quad \ y(t) =8t-t^{3}\), where the tangent line is horizontal or where it is vertical.
2. Show that \(C\) has two tangent lines at the point \(( 16,0)\).
3. Find equations of these tangent lines.
4. Graph \(C\).

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Arc Length

1. Find the arc length of one arch of the four-cusped hypocycloid \(x(t) =b\sin ^{3}t,\quad y(t) =b\cos ^{3}t,\quad 0\leq t\leq \dfrac{\pi }{2}\).
2. Graph the portion of the curve for \(0\leq t\leq \dfrac{\pi }{2}\).

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Arc Length

1. Find the arc length of the spiral \(x(t) =t\cos t,\quad y(t) =t\sin t,\) \(0\leq t\leq \pi \).
2. Graph the portion of the curve for \(0\leq t\leq \pi \).

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\(x(t)=3t,\quad y(t)=t^{2}-3;\; 0\leq t\leq 2\)

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\(x(t)=t^{2},\quad y(t)=3t;\;0\leq t\leq 2\)

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\(x(t)=\dfrac{t^{2}}{2}+1,\quad \ y(t)=\dfrac{1}{3}(2t+3)^{3/2};\; 0\leq t\leq 2\)

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\(x(t)=a\cos t,\quad \ y(t)=a\sin t; \;a > 0, 0\leq t\leq \pi \)

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\(x(t)=\cos (2t),\quad \ y(t)=\sin ^{2}t;\; 0\leq t\leq \dfrac{\pi }{2}\)

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\(x(t)=\dfrac{1}{t},\quad \ y(t)=\ln t;\;1\leq t\leq 2\)

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\(x(t) =20t,\quad y(t) =-16t^{2}\)

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\(x(t) =t+\cos t,\quad y(t) =2t-\sin t\)

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\(x(t) =20\sin ( 2t),\quad y(t) =6\cos t\)

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Arc Length along a Circle Use the arc length formula for parametric equations to show that for a circle of radius \(r,\) the length \(s\) of the arc subtended by a central angle of \(\theta \) radians is \(s=r\theta .\)

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If the smooth curve \(C\), represented by the parametric equations \(x=x(t), y=y(t),\quad \ a\leq t\leq b,\) is the graph of a function \(y=f(x)\), then \(x\) can be used in place of the parameter \(t,\) and the parametric equations for \(C\) are \(x=t,\quad y=f(t),\quad a\leq t\leq b.\) Show that the arc length formula for these parametric equations takes the form \[ \begin{equation*} s=\int_{a}^{b}\sqrt{\left( \frac{dx}{dt}\right) ^{2}+\left( \frac{dy}{dt} \right) ^{2}}\,dt=\int_{a}^{b}\sqrt{1+[ f' ( x) ] ^{2}}\,dx \end{equation*} \]

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\(x(t) =t^{1/3},\;y(t) =t^{2}\) from \(t=1\) to \(t=1.1\)

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\(x(t) =\sqrt{t},\;y(t) =t^{3}\) from \(t=1\) to \(t=1.2\)

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\(x(t) =a\sin t,\;y(t) =b\cos t\) from \(t=0\) to \(t=0.1\)

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\(x(t) =e^{at},\;y(t) =e^{bt}\) from \(t=0\) to \(t=0.2\)

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Show that a smooth curve \(x=f(t),\;y=g(t)\), for which \( \dfrac{dx}{dt}\) is never \(0\), can be represented by a rectangular equation \( y=F(x),\) where \(F\) is differentiable. [Hint: Use the fact that \(t=f^{-1}( x) \) exists and is differentiable.]

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Find the point on the curve \(x=\dfrac{4}{3}t^{3}+3t^{2},\;y=t^{3}-4t^{2}\), for which the length of the curve from \((0,0)\) to \((x, y)\) is \(\dfrac{80\sqrt{2}-40}{3}\).

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Higher-Order Derivatives Find an expression for \(\dfrac{ d^{2}y}{dx^{2}}\) if \(x=f(t),\;y=g(t)\), where \(f\) and \(g\) have second-order derivatives, and \(\dfrac{dx}{dt}\) is never 0.

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Find \(\dfrac{d^{2}y}{dx^{2}}\) if \(x( \theta) =a \cos ^{3}\theta ,\;y( \theta) =a \sin ^{3}\theta \).

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Consider the cycloid defined by \(x(t)=a(t-\sin t),\;y(t)=a(1-\cos t),\;a>0\). Discuss the behavior of the tangent line to the cycloid at \(t=0\).